Arithmetic Power Geometry for Image Blending: Localized Static Potentials and Global Spectral-Green Optimization

Image blending requires insertion of a source object into a target canvas while reducing boundary discontinuity and maintaining internal gradient coherence. This paper presents two deterministic image blending methods derived from Arithmetic Power Geometry (APG). APG VII is formulated as a localized static-potential method that restricts correction to a narrow boundary band around the inserted object. APG VIII is formulated as a global Spectral–Green optimization method that represents the foreground mask as a sparse pixel graph, constructs the graph Laplacian L = D − W, and solves a Poisson/Laplacian system over the active object domain. The methods are evaluated on COCO 2017 validation images using instance segmentation masks for source-object extraction and controlled source–target pair generation. On 30 evaluated image-mask pairs, naive pasting obtained 23.83 dB PSNR and 0.9216 SSIM. APG VII improved the result to 28.90 dB PSNR and 0.9418 SSIM with 0.0006 s average runtime. APG VIII obtained 32.07 dB PSNR and 0.9639 SSIM with 0.0670 s average runtime. Paired Wilcoxon signed-rank tests showed that APG VIII improved over APG VII for PSNR (p = 0.00033) and SSIM (p = 0.00407). The results show that APG VII provides low-cost local seam correction, while APG VIII provides higher quantitative reconstruction quality by propagating boundary and gradient constraints through the full foreground graph.